The two equity classes of the standard formula, how they combine inside the equity sub-module, and why the 10-point gap mostly disappears at an insurer that already holds listed equity.
Under the Solvency II standard formula, type 1 equity takes a 39 per cent shock and type 2 equity 49 per cent, both before the symmetric adjustment, and the two are combined with a 0.75 correlation. On a standalone basis a 100 million private equity fund therefore costs 10 points more capital than listed equity. But at an illustrative insurer that already holds 400 million of listed equity, the fund adds 39.46 million, 39.5 per cent, against 39.0 per cent for another 100 million of listed shares.
Worked in full in Insurance Capital for Asset Managers by Julian R. Sterling, with every figure reproduced in a free workbook.See the book on Amazon →
Type 1 equities are mainly those listed in regulated markets in countries of the EEA or the OECD; since the 2019 review of the Delegated Regulation, units of closed-ended, unleveraged alternative investment funds established in the EU can also qualify. Type 2 is everything else that is charged as equity: shares listed only elsewhere, unlisted shares, private equity, hedge funds, commodities and other alternative investments not caught by another sub-module. Fund units are looked through where the insurer can; a fund that cannot be looked through falls to type 2 as a whole, the case worked in what a missing look-through pack costs. Specific treatments take lower shocks and are left aside here: 22 per cent for strategic participations and for qualifying long-term equity investments, and 30 per cent plus a share of the symmetric adjustment for qualifying infrastructure equity. The type 1 and type 2 shocks and the 0.75 correlation are set in Articles 168 and 169 of Delegated Regulation (EU) 2015/35.
The symmetric adjustment is added to both shocks alike, so it does not change the gap between them; it is set to zero here for clarity, and its own effect is in how the symmetric adjustment changes a Solvency II equity charge.
| Item | Value |
|---|---|
| Listed equity already held (type 1) | 400.0 |
| Private equity fund to add (type 2) | 100.0 |
| Type 1 shock | 39% |
| Type 2 shock | 49% |
| Symmetric adjustment | 0.0% |
| Correlation, type 1 with type 2 | 0.75 |
Type 1 charge = 400.0 × 39% = 156.0; type 2 charge = 100.0 × 49% = 49.0
Equity charge = √(156.0² + 2 × 0.75 × 156.0 × 49.0 + 49.0²) = √(24,336 + 11,466 + 2,401) = √38,203 = 195.46
Before the fund the equity charge was 156.00, so the marginal charge = 195.46 − 156.00 = 39.46
Diversification inside the sub-module = 156.0 + 49.0 − 195.46 = 9.54
Excel, with type 1 and type 2 charges in B1 and B2: =SQRT(B1^2+2*0.75*B1*B2+B2^2)
Adding 100.0 of listed equity instead takes the type 1 book to 500.0 and the charge to 195.00, a marginal 39.00. The private equity fund costs 0.46 more, half a point, not ten. The reason is the correlation: below 1, a second bucket diversifies against the first, and the larger the first bucket, the more of the type 2 charge is absorbed. For a very small type 2 addition the marginal rate tends to 0.75 × 156.0 / 156.00 × 49%, which is 36.8 per cent, cheaper than type 1.
| Listed equity held | Equity charge before | After PE fund | Marginal, PE fund | Marginal, listed | Gap, points |
|---|---|---|---|---|---|
| 0 | 0.00 | 49.00 | 49.0% | 39.0% | 10.0 |
| 100 | 39.00 | 82.39 | 43.4% | 39.0% | 4.4 |
| 200 | 78.00 | 119.24 | 41.2% | 39.0% | 2.2 |
| 400 | 156.00 | 195.46 | 39.5% | 39.0% | 0.5 |
| 800 | 312.00 | 350.25 | 38.3% | 39.0% | −0.7 |
For an insurer with no listed equity, the full 10-point gap applies. For one with 800 of listed equity, the private equity fund is cheaper at the margin than more listed shares. The direction reverses for an insurer already heavy in type 2:
| Type 2 already held | Equity charge before | After PE fund | Marginal charge | As % of fund |
|---|---|---|---|---|
| 0 | 156.00 | 195.46 | 39.46 | 39.5% |
| 100 | 195.46 | 238.48 | 43.02 | 43.0% |
| 300 | 283.45 | 329.57 | 46.12 | 46.1% |
Suppose, illustratively, the private equity fund is expected to return 9 per cent and listed equity 7 per cent. Standalone, the two look alike on capital: 18.4 per cent for private equity (9 over 49) and 17.9 per cent for listed (7 over 39). At the insurer with 400.0 of listed equity, on the marginal charge, private equity earns 9.0 on 39.46, 22.8 per cent, while listed stays at 17.9 per cent. The ranking an asset manager shows from the standalone grid is not the ranking the insurer's capital committee will see.
Pitching type 2 assets with the standalone 49 per cent and adding it to the type 1 charge as if the two were perfectly correlated. Summing gives 205.0 here; the formula gives 195.46. The opposite error is just as common: quoting the 39.5 per cent marginal figure to an insurer that holds no listed equity, for whom it is 49.0. These figures are also only the equity sub-module; the equity charge is then aggregated with interest rate, spread, property and the rest, which dilutes it further, as in how to calculate the marginal SCR of an investment.
Charge type 1 at 39 and type 2 at 49 per cent plus the symmetric adjustment, combine them at 0.75, and compute the change rather than the standalone figure. Beside 400.0 of listed equity, a 100.0 private equity fund costs 39.46, half a point more than listed equity, not ten. The standard formula grids and the marginal capital comparison for the book's four funds are in the free workbook for this case.
Type 2 covers equities listed only outside the EEA and OECD regulated markets, unlisted equities, most private equity held outside a qualifying closed-ended, unleveraged EU fund, hedge funds, commodities and other alternative investments that do not fall into another category. Its standard formula shock is 49 per cent before the symmetric adjustment, against 39 per cent for type 1, so 100 million of type 2 carries a standalone charge of 49.0 million.
With a 0.75 correlation: the equity charge is the square root of the type 1 charge squared, plus 1.5 times the product of the two charges, plus the type 2 charge squared. Charges of 156.0 and 49.0 million therefore combine to 195.46 million, not 205.0, a diversification benefit of 9.54 million.
No. The marginal charge depends on what the insurer already holds. For an insurer with no equity, 100 million in a private equity fund adds 49.00 million. With 400 million of listed equity it adds 39.46 million, and an insurer that already holds 300 million of type 2 pays 46.12 million for the same fund.
This article is one calculation from Insurance Capital for Asset Managers. The book takes the same case from first principles to the decision, chapter by chapter, and every figure it prints is a live formula in the free companion workbooks.
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